Variational problems for the system of nonlinear Schr\'odinger equations with derivative nonlinearities
- Resource Type
- Working Paper
- Authors
- Hirayama, Hiroyuki; Ikeda, Masahiro
- Source
- Subject
- Mathematics - Analysis of PDEs
Mathematical Physics
Mathematics - Functional Analysis
35Q55
- Language
We consider the Cauchy problem of the system of nonlinear Schr\"odinger equations with derivative nonlinearlity. This system was introduced by Colin-Colin (2004) as a model of laser-plasma interactions. We study existence of ground state solutions and the global well-posedness of this system by using the variational methods. We also consider the stability of traveling waves for this system. These problems are proposed by Colin-Colin as the open problems. We give a subset of the ground-states set which satisfies the condition of stability. In particular, we prove the stability of the set of traveling waves with small speed for $1$-dimension.
Comment: Introduction is modified and references are updated